It's been an incredibly busy week, so I'm just now getting around to writing about Chad's post about how it's not science without graphs. Basically, in a fit of procrastination, he plotted his latest blog traffic stats into a nice little graph, drew a line through the data points, and analyzed the results. It's all very meta of him. But who am I to point fingers? Chad's post made me realize that I am officially an uber-geek. See, back in late January, I got sidelined by the flu and spent a couple of days with a high, spiking fever, unable to do much except moan in between gulps of Theraflu. Bored with flipping channels and the meager offerings of daytime television, I started checking my temperature every hour and recording it, with the aim of plotting it onto a graph when I was done. I had some vague, drug-fogged notion of finding the slope of the tangent curve and thereby practicing my calculus by taking a derivative using a "real-world" example: the rate of change of my body temperature as the fever ran its wicked course.
It didn't quite work out that way: that particular calculus trick only works if the graph gives you a smooth curve. I had so few data points that the result was a series of spiked lines. If I took my temperature every 5 minutes and plotted it out, the end result might have been closer to a curve -- or not. Given the relative crudeness of my digital thermometer, the differences at that point would be so minimal that it probably would have just looked like a straight line. Still, before I started my amateur dabbling into self-taught calculus, I would not have realized that the closer one gets to an infinite number of ever-smaller data points, the more like a curve the resulting graphed data will appear. And it would never have occurred to me to try to create my own real-world calculus problem tracking the rate of change of my own body temperature. Maybe I ended up somewhere other than where I'd intended when I started my little sickbed exercise, but I learned something quite valuable from the experience -- and I'm not likely to forget the "lesson," either.
Real-world examples while learning abstract mathematical principles work for me, despite the recent findings by researchers at Ohio State University that this widespread assumption among educators may be wrong. Ed Yong at Not Exactly Rocket Science has an excellent summation of the study specifics, accompanied by a thought-provoking comment thread. For instance, more than one person said that the so-called "real world" problems one finds in, say, calculus textbooks bear very little resemblance to anything most students would want to solve -- like that silly train analogy that leads off both the New York Times article and Ed's blog post on the study's results. (Jen-Luc Piquant has her own snide response to when Train A, departing at 6 PM and traveling at 40 MPH toward Station B, will pass Train B, departing at 7 PM and traveling at 50 MPH toward Station A: "When everyone on board is long past caring.") Far from making math "come alive," it's just one more way to make students' eyes glaze over in boredom.
I do not, however, conclude from this that "real world examples don't work." I think it depends on which examples you choose, and how you use them. They are a useful starting point for piquing student interest, but you still have to make the critical connection -- "This relates to that abstract principle, which can be broadly applied to other situations" -- and put in the work to grasp the abstractions.
Jennifer Kaminski, the OSU researcher who spear-headed the study, thinks such an approach obscures the underlying mathematical principle, rather than illuminating it, and actually hinders students' ability to transfer their knowledge to new problems. "They tend to remember the superficial, two trains passing in the night," she told the New York Times. "It's really a problem of our attention getting pulled to superficial information." I can see how that might happen, but I think it's more of a translation problem. Honestly? I sucked at textbook story problems in my K-12 math classes, and received excellent grades in high school geometry and algebra.
But here's the thing: I didn't actually understand the abstractions; I was just blindly following the "rules," manipulating meaningless symbols. And it bored me. I needed some kind of context, just not the equally pointless exercises routinely used in classrooms. The real world examples in textbooks don't really correspond to our daily experiences, or how we might typically approach such a problem. As one of Ed's commenters put it: "If I wanted to know Frankie's and Johnny's ages, I'd ask them, not work out some weird algebra problem." Yet another commenter observed, "'Real-world examples may be treated by students as confusing symbolic concepts that look like real things they know about but act like abstract notions that are defined by the teacher."
I was pleased to read that Kaminski isn't suggesting that we eliminate all real-world examples in classrooms; rather, she thinks that they should augment the abstract principles -- which should be taught first -- rather than being deeply grounded in one specific context. I agree this might increase a student's chances of extrapolating the general principles and applying them to new problems as they arise. Perhaps letting the students choose a real-world problem they'd like to solve -- like my little experiment plotting out my changing rate of body temperature -- is a better way of incorporating a practical context.
You're more likely to pique their interest if they're involved in creating the problems and then figuring out how to solve them -- the "lessons" they learn along the way are more likely to "stick," plus it's a lot more similar to what a working scientist actually does for a living. Is it a calculus problem? A statistical one? How does one go about "translating" that situation into a meaningful mathematical format? This is more of a ground-up approach, akin to taking apart an alarm clock and putting it back together to gain a more comprehensive understanding of how it works. Personally, Jen-Luc would like to see more LOLCats in math and science classes:
This kind of choose-your-own-problems approach also might address the perennial problem of over-generalization -- we all learn differently, and suggesting there is only one correct way to teach a subject like math or physics is likely to leave behind as many students as such a pedagogical approach would advance. And sometimes teachers under-estimate the difficult of new concepts because it's been so long since they learned the material for the first time themselves. As commenter Sam C. said, "Once one has learned something, it's difficult to appreciate what it looks like to someone who hasn't learned it." Something that seems perfectly obvious to the teacher, probably needs to be spelled out, step by step, for many of his/her students.
Case in point: I started my informal calculus "studies" with a DVD lecture series from The Teaching Company. The lectures were pretty good, conceptually: visual elements, real-world examples, but tying them to the abstract principles and then showing how they could be broadly applied. The first thing I learned was how I could (a) use the derivative to figure out the speed of my car from the car's position, and (b) use the integral to figure out how far I'd traveled in my car based on speed. The two are flip sides of the same coin, two different approaches to solving the same problem, depending on the information at one's disposal. And there's a handy real-world context: this is basically what's going on in your car's speedometer and odometer all the time.
Frankly, finding the integral is a labor-intensive process of multiplication and addition take to ridiculous extremes (i.e., infinity). There is a short-cut to the much-harder integral however: if I know both my beginning and ending position, for example, I could just subtract the first from the second to figure out how far I'd traveled. What if I don't know my ending position (and my odometer is broken), just my speed (the velocity function)? Per my DVD instructor, "all" I have to do is figure out which position function generates the known velocity function, and voila! I can do a bit of math-y hocus-pocus to essentially "retrace my steps" backward and use the easier derivative approach. Fair enough, but he never once explained how one goes about finding that position function. There's a lot of them. Still, he insisted it was a simple matter, and silly me -- I believed him.
My DVD instructor lied. It's actually a non-trivial thing for someone just starting out, and/or a bit rusty in their basic algebra and geometry. Don't take my word for it; listen to Johann Bernoulli, a contemporary of Newton and Leibniz who made significant contributions to then-brand-new field of calculus in the 17th century: "But just as much as it is easy to find the differential (derivative) of a given quantity, so it is difficult to find the integral of a given differential," he wrote. "Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not."
Fortunately, I know a lot of physicists and a smattering of mathematicians, most of whom are happy to weigh in now and then with their own insights and "tricks" for the kind of road-block described above. And I'm persistent. I only bring it up because I think it's always interesting to see where different people get hung up when learning new mathematical concepts. Sometimes it's just a language problem, mixing up terminology, or not realizing that you do know what a particular term means -- you just didn't realize that's what your mental concept was called. You hadn't made the connection. Sometimes the instructor has inadvertently left out a step, or doesn't realize that some of his/her students need to be walked through something a bit more carefully.
Because we all learn and think differently -- newsflash: even scientists don't all think and learn alike! -- I'm interested in hearing from readers about similar experiences in their math and science education -- or even their humanities education. I admit, I have an "intuitive" feel for words and writing, and have been guilty in the past of just not understanding why someone couldn't grasp some "trivial" aspect of composition. I've noticed that many "gifted" math sorts can make similar intuitive leaps with numbers. What were your most significant roadblocks? Have you ever stopped to really analyze what happened? How did you overcome them? What are some of the "tricks of the trade" you find useful when applying abstract math principles to "real world" problems?
I think it's a conversation worth having....